By [1]}, we see that \(\varrho \in \sigma ({B}^*)\)  (actually considering the complexification of \({B}^*\)  and \(Y_\infty \) )
if and only if \(\left(\frac{1}{|\varrho |}B_n\right)_{n\in \mathbb {Z}}\)  does not admit a strong exponential dichotomy with respect to the sequence of norms \(\mathcal {S}^{\prime }\) . This fact shows (see Definition REF  and Remark REF ) that
\(|\sigma ({B}^*)|=\Sigma _{ED, \mathbb {B}, \mathcal {S}^{\prime }},\) 
